Who exactly handles this? Is this done by the US Government via one of its printing offices? Or is this private? I know Jeppesen does some charts, especially approach plates, but I am not totally sure how all this is done.
I believe the FAA got out of the printing business by implementing a new business model for charts where they license the product to private industry who then take on the risk/reward of the printing process for their customers.
From the FAA AIS website:
The Federal Aviation Administration (FAA), Aeronautical Information Services (AJV-5), is transitioning aeronautical charts and related products print and distribution operations from existing contract(s) to multiple no-cost agreements with private industry. Digital product files allow industry the opportunity to print, sell, and distribute aeronautical charting products to the public in an Available on Demand (AOD) approach. Any party interested in the opportunity to become an approved print provider should submit product qualification samples from the files below (in accordance with instructions in the Print Provider agreement).
More detailed information about the program is also available through the FAA contract award management site via an official announcement. So to answer your question, the printing of charts is now handled by private industry, not the FAA.
Since you mentioned Jeppesen specifically, note that they are a bit of a different concept because they take raw data from each state's official AIP and then integrate it together in their own way and perform other value-added services. They then print their own end-product for their customers. It is based on the same raw data but their product should not be considered the same as what is produced by the FAA.
The art and science of creating maps is called Cartography. The experts in the field, Cartographers, are educated for the same with degrees in Geography. There are many schools that offer such programs.
The FAA, which falls under the DOT (Depart of Transportation), carries out a tonne of duties. "Air Traffic" is one of them, and creating & publishing charts are duties of the offices that handle the Air Traffic section.
That said, one of the offices[a] under the Air Traffic Sector under the FAA has a department that employs Cartographers and other agents to create and work on these maps. How big is the department, and how do they "actually" create these maps (what software and machines they use, how are they supervised etc.) are questions that can't be answered.
As far is printing and publication is concerned, the previous answer covers it.